Supertasks

First published Tue Apr 5, 2016

[Editor's Note: The following new entry by John Manchak and Bryan W. Roberts replaces the
former entry
on this topic by the previous author.]

A supertask is a task that consists in infinitely many component
steps, but which in some sense is completed in a finite amount of
time. Supertasks were studied by the pre-Socratics and continue to be
objects of interest to modern philosophers, logicians and physicists.
The term “super-task” itself was coined by J.F. Thomson
(1954).

Here we begin with an overview of the analysis of supertasks and their
mechanics. We then discuss the possibility of supertasks from the
perspective of general relativity.

1. Mechanical properties

Strange things can happen when one carries out an infinite task.

For example, consider a hotel with a countably infinite number of
rooms. One night when the hotel is completely occupied, a traveler
shows up and asks for a room. “No problem,” the
receptionist replies, “there’s plenty of space!” The first
occupant then moves to the second room, the second to the third room,
the third to the fourth room, and so on all the way up. The result is
a hotel that has gone from being completely occupied to having one
room free, and the traveler can stay the night after all. This
supertask was described in a 1924 lecture by David Hilbert, as
reported by Gamow (1947).

One might take such unusual results as evidence against the
possibility of supertasks. Alternatively, we might take them to seem
strange because our intuitions are based on experience with finite
tasks, and which break down in the analysis of supertasks. For now,
let us simply try to come to grips with some of the unusual mechanical
properties that supertasks can have.

1.1 Missing final and initial steps: The Zeno walk

Supertasks often lack a final or initial step. A famous example is the
first of
Zeno’s Paradoxes,
the Paradox of the Dichotomy. The runner Achilles begins at the
starting line of a track and runs ½ of the distance to the
finish line. He then runs half of the remaining distance, or ¼
of the total. He then runs half the remaining distance again, or
&frac18; of the total. And he continues in this way ad infinitum,
getting ever-closer to the finish line (Figure 1.1.1). But there is no
final step in this task.

Fig 1.1.1. The Zeno Dichotomy supertask.

There is also a “regressive” version of the Dichotomy
supertask that has no initial step. Suppose that Achilles does reach
the finish line. Then he would have had to travel the last ½ of
the track, and before that ¼ of the track, and before that
&frac18; of the track, and so on. In this description of the Achilles
race, we imagine winding time backwards and viewing Achilles getting
ever-closer to the starting line (Figure 1.1.2). But now there is no
initial step in the task.

Fig 1.1.2. Regressive version of the Zeno Dichotomy supertask.

Zeno, at least as portrayed in Aristotle’s Physics, argued
that as a consequence, motion does not exist. Since an infinite number
of steps cannot be completed, Achilles will never reach the finish
line (or never have started in the regressive version). However,
modern mathematics provides ways of explaining how Achilles can
complete this supertask. As Salmon (1998) has pointed out, much of the
mystery of Zeno’s walk is dissolved given the modern definition of a
limit. This provides a precise sense in which the following sum
converges:

Although it has infinitely many terms, this sum is a geometric series
that converges to 1 in the standard topology of the real numbers. A
discussion of the philosophy underpinning this fact can be found in
Salmon (1998), and the mathematics of convergence in any real analysis
textbook that deals with infinite series. From this perspective,
Achilles actually does complete all of the supertask steps in the
limit as the number of steps goes to infinity. One might only doubt
whether or not the standard topology of the real numbers provides the
appropriate notion of convergence in this supertask. A discussion of
the subtleties of the choice of topology has been given by Mclaughlin
(1998).

Max Black (1950) argued that it is nevertheless impossible to complete
the Zeno task, since there is no final step in the infinite sequence.
The existence of a final step was similarly demanded on a priori terms
by Gwiazda (2012). But as Thomson (1954) and Earman and Norton (1996)
have pointed out, there is a sense in which this objection equivocates
on two different meanings of the word “complete.” On the
one hand “complete” can refer to the execution of a final
action. This sense of completion does not occur in Zeno’s Dichotomy,
since for every step in the task there is another step that happens
later. On the other hand, “complete” can refer to carrying
out every step in the task, which certainly does occur in Zeno’s
Dichotomy. From Black’s argument one can see that the Zeno Dichotomy
cannot be completed in the first sense. But it can be completed in the
second. The two meanings for the word “complete” happen to
be equivalent for finite tasks, where most of our intuitions about
tasks are developed. But they are not equivalent when it comes to
supertasks.

Hermann Weyl (1949, §2.7) suggested that if one admits that the
Zeno race is possible, then one should equally admit that it is
possible for a machine to carry out an infinite number of tasks in
finite time. However, one difference between the Zeno run and a
machine is that the Zeno run is continuous, while the tasks carried
out by a machine are typically discrete. This led Grünbaum (1969)
to consider the “staccato” version of the Zeno run, in
which Achilles pauses for successively shorter times at each
interval.

1.2 Missing limits: Thomson’s Lamp

Supertasks are often described by sequences that do not converge. J.
F. Thomson (1954) introduced one such example now known as Thomson’s
Lamp, which he thought illustrated a sense in which supertasks truly
are paradoxical.

Suppose we switch off a lamp. After 1 minute we switch it on. After
½ a minute more we switch it off again, ¼ on, &frac18;
off, and so on. Summing each of these times gives rise to an infinite
geometric series that converges to 2 minutes, after which time the
entire supertask has been completed. But when 2 minutes is up, is the
lamp on or off?

Fig 1.2.1. Thomson’s lamp.

It may seem absurd to claim that it is on: for each moment that the
lamp was turned on, there is a later moment at which it was turned
off. But it would seem equally absurd to claim that it is off: for
each moment that the lamp is turned off, there is a later moment that
it was turned on. This paradox, according to Thomson, suggests that
the supertask associated with the lamp is impossible.

To analyze the paradox, Thomson suggested we represent the
“on” state of the map with the number 1 and the
“off” state with 0. The supertask then consists in the
sequence of states,

\[
0, 1, 0, 1, 0, 1, \ldots .
\]

This sequence does not converge to any real number in the standard
real topology. However, one might redefine what it means for a
sequence to converge in response to this. For example, we could define
convergence in terms of the arithmetic mean. Given a sequence
\(x_n\), the Cesàro mean is the sequence
\(C_1 = x_1\), \(C_2 = (x_1 + x_2)/2\), \(C_3 = (x_1 + x_2 +
x_3)/3\), and so on. These numbers describe the average value
of the sequence up to a given term. One says that a sequence
\(x_n\) Cesàro converges to a number \(C\) if and
only if \(C_n\) converges (in the ordinary sense) to \(C\). It is
then well-known that the sequence \(0, 1, 0, 1, \ldots\) Cesàro
converges to ½ (see e.g. Bashirov 2014).

Thomson pointed out that this argument is not very helpful without an
interpretation of what lamp-state is represented by ½. We want
to know if the lamp is on or off; saying that its end state is
associated with a convergent arithmetic mean of ½ does little
to answer the question. However, this approach to resolving the
paradox has still been pursued, for example by Pérez
Laraudogoita, Bridger and Alper (2002) and by Dolev (2007).

Are there other consistent ways to describe the final state of
Thomson’s lamp in spite of the missing limit?

Benacerraf (1962) pointed out a sense in which the answer is yes. The
description of the Thomson lamp only actually specifies what the lamp
is doing at each finite stage before 2 minutes. It says nothing about
what happens at 2 minutes, especially given the lack of a converging
limit. It may still be possible to “complete” the
description of Thomson’s lamp in a way that leads it to be either on
after 2 minutes or off after 2 minutes. The price is that the final
state will not be reached from the previous states by a convergent
sequence. But this by itself does not amount to a logical
inconsistency.

Such a completion of Thomson’s description was explicitly constructed
by Earman and Norton (1996) using the following example of a bouncing
ball.

Suppose a metal ball bounces on a conductive plate, bouncing a little
lower each time until it comes to a rest on the plate. Suppose the
bounces follow the same geometric pattern as before. Namely, the ball
is in the air for 1 minute after the first bounce, ½ minute
after the second bounce, ¼ minute after the third, &frac18;
minute after the fourth, and so on. Then the entire infinite sequence
of bounces is a supertask.

Now suppose that the ball completes a circuit when it strikes the
metal plate, thereby switching on a lamp. This is a physical system
that implements Thomson’s lamp. In particular, the lamp is switched on
and off infinitely many times over the course of a finite duration of
2 minutes.

Fig 1.2.2.
Thomson’s lamp implemented by a bouncing ball: contact of
the bouncing ball with the plate switches the Thomson lamp on. The
supertask ends with the lamp on.

What is the state of this lamp after 2 minutes? The ball will have
come to rest on the plate, and so the lamp will be on. There is no
mystery in this description of Thomson’s lamp.

Alternatively, we could arrange the ball so as to break the circuit
when it makes contact with the plate. This gives rise to another
implementation of Thomson’s lamp, but one that is off after 2 minutes
when the ball comes to its final resting state.

Fig 1.2.3.
Another implementation of Thomson’s lamp: contact of the bouncing ball
with the plate switches the Thomson lamp off. The supertask ends with
the lamp off.

These examples show that is possible to fill in the details of
Thomson’s lamp in a way that either renders it definitely on after the
supertask, or definitely off. For this reason, Earman and Norton
conclude with Benacerraf that the Thomson lamp is not a matter of
paradox but of an incomplete description.

As with the Zeno Dichotomy, there is a regressive version of the
Thomson lamp supertask. Such a lamp has been studied by Uzquiano
(2012), although as a set of instructions rather than a set of tasks.
Consider a lamp that has been switched on at 2 seconds
past the hour, off at 1 second past, on at ½ a second past, off
at ¼ a second past, and so on. What is the state
of the lamp on the hour, just before the supertask has begun? This
supertask can be viewed as incomplete in the same way as the original
Thomson lamp. Insofar as the mechanics of bouncing balls and electric
circuits described in Earman and Norton’s lamp are time reversal invariant, it follows that the time-reversed system is a possibility as well, which is spontaneously excited to begin bouncing, providing a physical implementation of the regressive Thomson
lamp. However, whether the reversed Thomson lamp is a physical possibility
depends on whether or not the system is time reversible. A difficulty is that its initial state will not determine the subsequent history of an infinity of alternations.

1.3 Discontinuous quantities: The Littlewood-Ross Paradox

Sometimes supertasks require a physical quantity to be discontinuous
in time. One example of this, known as Ross’ paradox, was described by
John Littlewood (1953) as an “infinity paradox” and
expanded upon by Sheldon Ross (1988) in his well-known textbook on
probability. It goes as follows.

Suppose we have a jar—a very large jar—with the
capacity to hold infinitely many balls. We also have a countably
infinite pile of balls, numbered 1, 2, 3, 4, …. First we drop balls
1–10 into the jar, then remove ball 1. (This adds a total of nine
balls to the jar.) Then we drop balls 11–20 in the jar, and remove
ball 2. (This brings the total up to eighteen.) Suppose that we
continue in this way ad infinitum, and that we do so with
ever-increasing speed, so that we will have used up our entire
infinite pile of balls in finite time (Figure 1.3.1). How many balls
will be in the jar when this supertask is over?

Fig 1.3.1.
The Littlewood-Ross procedure.

Both Littlewood (1953) and Ross (1976) responded that the answer is
zero. Their reasoning went as follows.

Ball 1 was removed at the first stage. Ball 2 was removed at the
second stage. Ball n was removed at the nth stage, and so on ad
infinitum. Since each ball has a label n, and since each label n was
removed at the nth stage of the supertask, there can be only be zero
balls left in the jar at the end after every stage has been completed.
One can even identify the moment at which each of them was
removed.

Some may be tempted to object that, on the contrary, the number of
balls in the jar should be infinite when the supertask is complete.
After the first stage there are 9 balls in the jar. After the second
stage there are 18. After the third stage there are 27. In the limit
as the number of stages approaches infinity, the total number of balls
in the jar diverges to infinity. If the final state of the jar is
determined by what the finite-stage states are converging to, then the
supertask should conclude with infinitely many balls in the jar.

If both of these responses are equally reasonable, then we have a
contradiction. There cannot be both zero and infinity balls in a jar.
It is in this sense that the Littlewood-Ross example might be a
paradox.

Allis and Koetsier (1991) argued that only the first response is
justified because of a reasonable “principle of
continuity”: that the positions of the balls in space are a
continuous function of time. Without such a principle, the positions
of the balls outside the jar could be allowed to teleport
discontinuously back into the jar as soon as the supertask is
complete. But with such a principle in place, one can conclude that
the jar must be empty at the end of the supertask. This principle has
been challenged by Van Bendegum (1994), with a clarifying rejoinder by
Allis and Koetsier (1996).

Earman and Norton (1996) follow Allis and Koetsier (and Littlewood and
Ross) in demanding that the worldlines of the balls in the jar be
continuous, but point out that there is a different sense of
discontinuity that develops as a consequence. (A
‘worldline’ is used here to describe the trajectory of a
particle through space and time; it is discussed more below in the
section on Time in Relativistic Spacetime.)
Namely, if one views the number of balls in the jar as approximated by
a function \(N(t)\) of time, then this “number
function” is discontinuous in the Littlewood-Ross supertask,
blowing up to an arbitrarily large value over the course of the
supertask before dropping discontinuously to 0 once it is over. In
this sense, the Littlewood-Ross paradox presents us with a choice, to
either,

Take the worldline of each ball in the jar to be continuous in
time; or

Take the number \(N(t)\) of balls in the jar to be approximated by a
continuous function of time;

but not both. The example thus seems to require a physical quantity to
be discontinuous in time: either in the worldlines of the balls, or in
the number of balls in the jar.

A variation of the Littlewood-Ross example has been posed as a puzzle
for decision theory by Barrett and Arntzenius (1999, 2002). They
propose a game involving an infinite number of $1 bills, each numbered
by a serial number 1, 2, 3, …, and in which a person begins with $0.
The person must then choose between the following two options.

Option A: accept $1; or

Option B: first accept $2n+1, where n is the
number of times the offer has been made, and then return whatever bill
the player holds with the smallest serial number.

At each finite stage of the game it appears to be rational to choose
Option B. For example, at stage n=1 Option B returns $3, while
Option A returns $1. At stage n=2 Option B returns $7 while
Option A returns $1. And so on.

However, suppose that one plays this game as a supertask, so that the
entire infinite number of offers is played in finite time. Then how
much money will the player have? Following exactly the same reasoning
as in the Littlewood-Ross paradox, we find that the answer is $0. For
each bill’s serial number, there is a stage at which that bill was
returned. So, if we presume the worldlines of the bills must be
continuous, then the infinite game ends with the player winning
nothing at all. This is a game in which the rational strategy at each
finite stage does not provide a winning strategy for the infinite
game.

There are variations on this example that have a more positive yield
for the players. For example, Earman and Norton (1996) propose the
following pyramid marketing scheme. Suppose that an agent sells two
shares of a business for $1,000 each to a pair of agents. Each agent
splits their share in two and sells it for $2,000 to two more agents,
thus netting $1,000 while four new agents go into debt for $1,000
each. Each of the four new agents then do the same, and so on ad
infinitum. How does this game end?

If the pool of agents is only finitely large, then the last agents
will get saddled with the debt while all the previous agents make a
profit. But if the pool is infinitely large, and the pyramid marketing
scheme becomes a supertask, then all of the agents will have profited
when it is completed. At each stage in which a given agent is in debt,
there is a later stage in which the agent sells to shares and makes
$1,000. This is thus a game that starts with equal total amount of
profit and debt, but concludes having converted the debt into pure
profit.

1.4 Classical mechanical supertasks

The discussions of supertasks so far suggest that the possibility of
supertasks is not so much a matter of logical possibility as it is
“physical possibility.” But what does “physical
possibility” mean? One natural interpretation is that it means,
“possible according to some laws of physics.” Thus, we can
make the question of whether supertasks are possible more precise by
asking, for example, whether supertasks compatible with the laws of
classical particle mechanics.

Earman and Norton’s (1996) bouncing ball provides one indication that
the answer is yes. Another particularly simple example was introduced
by Pérez Laraudogoita (1996, 1998), which goes as follows.

Suppose an infinite lattice of particles of the same mass are arranged
so that there is a distance of ½ between the first and the
second, a distance of ¼ between the second and the third, a
distance of &frac18; between the third and the fourth, and so on. Now
imagine that a new particle of the same mass collides with the first
particle in the lattice, as in Figure 1.4.1. If it is a perfectly
elastic collision, then the incoming particle will come to rest and
the velocity will be transferred to the struck particle. Suppose it
takes ½ of a second for the second collision to occur. Then it
will take ¼ of a second for the third to occur, &frac18; of a
second for the fourth, and so on. The entire infinite process will
thus be completed after 1 second.

Fig 1.4.1.
Jon Pérez Laraudogoita’s ‘Beautiful Supertask’

Earman and Norton (1998) observed several curious facts about this
system. First, unlike Thomson’s lamp, this supertask does not require
unbounded speeds. The total velocity of the system is never any more
than the velocity of the original moving particle. Second, this
supertask takes place in a bounded region of space. So, there are no
boundary conditions “at infinity” that can rule out the
supertask. Third, although energy is conserved in each local
collision, the global energy of this system is not conserved, since
after finite time it becomes a lattice of infinitely many particles
all at rest. Finally, the supertask depends crucially on there being
an infinite number of particles, and the width of these particles must
shrink without bound while keeping the mass fixed. This means the mass
density of the particles must grow without bound. The failure of
global energy conservation and other curious features of this system
have been studied by Atkinson (2007, 2008), Atkinson and Johnson
(2009, 2010) and by Peijnenburg and Atkinson (2008) and Atkinson and
Peijnenburg (2014).

Another kind of classical mechanical supertask was described by
Pérez Laraudogoita (1997). Consider again the infinite lattice
of particles of the same mass, but this time suppose that the first
particle is motionless, that the second particle is headed towards the
first with some velocity, and that the velocity of each successive
particle doubles (Figure 1.4.2). The first collision sets the first
particle in motion. But a later collision then sets it moving faster,
and a later collision even faster, and so on.

Fig 1.4.2.
A supertask that relies on unbounded speed.

It is not hard to arrange this situation so that the first collision
happens after ½ of a second, the second collision after
¼ of a second, the third after &frac18; of a second, and so on
(Pérez Laraudogoita 1997). So again we have a supertask that is
completed after one second.

What is the result of this supertask? Their answer is that none of the
particles remain in space. They cannot be anywhere in space, since for
each horizontal position that a given particle can occupy there is a
time before 1 second that it is pushed out of that position by a
collision. The worldline of any one of the particles from this
supertask can be illustrated using Figure 1.4.3. This is what Malament
(2008, 2009) has referred to as a “space evader”
trajectory. The time-reversed “space invader” trajectory
is one in which the vacuum is spontaneously populated with particles
after some fixed time.

Fig 1.4.3. Worldline of the supertask particle.

Earman and Norton (1998) gave some variations on this supertask,
including one which occurs in a bounded region in space. Unlike the
example of Pérez Laraudogoita (1996), this supertask also
essentially requires particles to be accelerated to arbitrarily high
speeds, and in this sense is essentially non-relativistic. See
Pérez Laraudogoita (1999) for a rejoinder.

This supertask is modeled on an example of Benardete (1964), who
considered a space ship that successively doubles its speed until it
escapes to spatial infinity. Supertasks of this kind were also studied
by physicists like Lanford (1975, §4), who identified a system of
particles colliding elastically that can undergo an infinite number of
collisions in finite time. Mather and McGehee (1975) pointed out a
similar example. Earman (1986) discussed the curious behavior of
Lanford’s example as well, pointing out that such supertasks provide
examples of classical indeterminism, but can be eliminated by
restricting to finitely many particles or by imposing appropriate
boundary conditions.

1.5 Quantum mechanical supertasks

It is possible to carry some of the above considerations of supertasks
over from classical to quantum mechanics. The examples of quantum
mechanical supertasks that have been given so far are somewhat less
straightforward than the classical supertasks above. However, they
also bear a more interesting possible relationship to physical
experiments.

Example 1: Norton’s Lattice

Norton (1999) investigated whether there exists a direct quantum
mechanical analogue of the kinds of supertasks discussed above. He
began by considering the classical scenario shown in Figure 1.5.1 of
an infinite lattice of interacting harmonic oscillators. Assuming the
springs all have the same tension and solving the equation of motion
for this system, Norton found that it can spontaneously excite,
producing an infinite succession of oscillations in the lattice in a
finite amount of time.

Fig 1.5.1. Norton’s infinite harmonic oscillator system.

Using this example as a model, Norton produced a similar supertask for
a quantum lattice of harmonic oscillators. Begin with an infinite
lattice of 2-dimensional quantum systems, each with a ground state
\(\ket{\phi}\) and an excited state \(\ket{\chi}\). Consider the
collection of vectors,

For simplicity, we restrict attention to the possible states of the
system that are spanned by this set. We posit a Hamiltonian that has
the effect of leaving |0〉 invariant; of creating |1〉 and
destroying |2〉; of creating |2〉 and destroying |3〉; and
so on. Norton then solved the differential form of the
Schrödinger equation for this interaction and argued that it
admits solutions in which all of the nodes in the infinite lattice
start in their ground state, but all become spontaneously excited in
finite time.

Norton’s quantum supertask requires a non-standard quantum system
because the dynamical evolution he proposes is not unitary, even
though it obeys a differential equation in wavefunction space that
takes the form of the Schrödinger equation (Norton 1999,
§5). Nevertheless, Norton’s quantum supertask has fruitfully
appeared in physical applications, having been found to arise
naturally in a framework for perturbative quantum field theory
proposed by Duncan and Niedermaier (2013, Appendix B).

Example 2: Hepp Measurement

Although quantum systems may sometimes be in a pure superposition of
measurable states, we never observe our measurement devices to be in
such states when they interact with quantum systems. On the contrary, our measurement devices always seem to display definite values. Why? Hepp (1972)
proposed to explain this by modeling the measurement process using a
quantum supertask. This example was popularized by Bell (1987,
§6) and proposed as a solution to the measurement problem by Wan
(1980) and Bub (1988).

Here is a toy example illustating the idea. Suppose we model an
idealised measuring device as consisting in an infinite number of
fermions. We imagine that the fermions do not interact with each
other, but that a finite number of them will couple to our target
system whenever we make a measurement. Then an observable
characterising the possible outcomes of a given measurement will be a
product corresponding to some finite number n of observables,

Restricting to a finite number of fermions at a time has the effect of
splitting the Hilbert space of states into special subspaces called
superselection sectors, which have the property that when \(\ket{\psi}\)
and \(\ket{\phi}\) come from different sectors, any superposition
\(a\ket{\psi} + b\ket{\phi}\) with \(|a|^2 + |b|^2 = 1\) will be a mixed state. It turns out in
particular that the space describing the state in which all the
fermions are \(z\)-spin-up is in a different superselection sector than
the space in which they are all spin down. Although this may be puzzling
for the newcomer, it can be found in any textbook that deals with
superselection. And it allows us to construct an interesting supertask
describing the measurement process. The following simplified version
of it was given by Bell (1987).

Suppose we wish to measure a single fermion. We model this as a
wavefunction that zips by the locations of each fermion in our
measurement device, interacting locally with the individual fermions
in the device as it goes (Figure 1.5.2). The interaction is set up in
such a way that every fermion is passed in finite time, and such that
after the process is completed, the measurement device indicates what
the original state of the fermion being measured was. In particular,
suppose the single fermion begins in a \(z\)-spin-up state. Then,
after it has zipped by each of the infinite fermions, they will all be
found in the \(z\)-spin-up state. If the single fermion begins
in a \(z\)-spin-down state, then the infinite collection of
fermions would all be \(z\)-spin-down. What if the single fermion
was in a superposition? Then the infinite collection of fermions would
contain some mixture of \(z\)-spin up and \(z\)-spin down
states.

Fig 1.5.2. Bell’s implementation of the Hepp measurement supertask.

Hepp found that, because of the superselection structure of this
system, this measurement device admits mixed states that can indicate
the original state of the single fermion, even when the latter begins
in a pure superposition. Suppose we denote the \(z\)-spin
observable for the nth fermion in the measurement device as, \(s_n = I
\otimes I \otimes \cdots (n\,times) \cdots \otimes \sigma_z \otimes I
\cdots.\) We now construct a new observable, given by,

This observable has the property that \(\langle \psi, S\phi\rangle =
1\) if \(\ket{\psi}\) and \(\ket{\phi}\) both lie in the same
superselection sector as the state in which all the fermions in the
measurement device are \(z\)-spin-up. It also has the property
that \(\langle\psi,S\phi\rangle = -1\) if they lie in the same
superselection sector as the all-down state. But more interestingly,
suppose the target fermion that we want to measure is in a pure
superposition of \(z\)-spin-up and \(z\)-spin-down
states. Then, after it zips by all the fermions in the measurement
device, that measurement device will be left in a superposition of the
form \(a\ket{\uparrow} + b\ket{\downarrow}\), where \(\ket{\uparrow}\)
is the state in which all the fermions in the device are spin-up and
\(\ket{\downarrow}\) is the state in which they are all spin
down. Since \(\ket{\uparrow}\) and \(\ket{\downarrow}\) are in
different superselection sectors, it follows that their superposition
must be a mixed state. In other words, this model allows the
measurement device to indicate the pure state of the target fermion,
even when that state is a pure superposition, without the device
itself being in a pure superposition.

The supertask underpinning this model requires an infinite number of
interactions. As Hepp and Bell described it, the model was unrealistic
because it required an infinite amount of time. However, a similar
system was shown by Wan (1980) and Bub (1988) to take place in finite
time. Their approach appears at first glance to be a promising model
of measurement. However, Landsman (1991) pointed out that it is
inadequate on one of two levels: either the dynamics is not
automorphic (which is the analogue of unitarity for such systems), or
the task is not completed in finite time. Landsman (1995) has argued
that neither of these two outcomes is plausible for a realistic local
description of a quantum system.

Example 3: Continuous Measurement

Another quantum supertask is found in the so-called Quantum Zeno
Effect. This literature begins with a question: what would happen if
we were to continually monitor a quantum system, like an unstable
atom? The predicted effect is that the system would not change, even
if it is an unstable atom that would otherwise quickly decay.

Misra and Sudarshan (1977) proposed to make the concept of
“continual monitoring” precise using a Zeno-like
supertask. Imagine that an unstable atom is evolving according to some
law of unitary evolution \(U_t\). Suppose we measure whether or
not the atom has decayed by following that regressive form of Zeno’s
Dichotomy above. Namely, we measure it at time \(t\), but also at time
\(t/2\), and before that at time \(t/4\), and at time \(t/8\), and so on. Let \(E\) be
a projection corresponding to the initial undecayed state of the
particle. Finding the atom undecayed at each stage in the supertask
then corresponds to the sequence,

\[
EU_tE,\; EU_{t/2}E,\; EU_{t/4}E,\; EU_{t/8}E,\ldots.
\]

Misra and Sudarshan use this sequence as a model for continuous
measurement, by supposing that the sequence above converges to an
operator \(T(t)=E\), and that it does so for all times \(t\) greater than or
equal to zero. The aim is for this to capture the claim that the atom
is continually monitored beginning at a fixed time \(t=0\). They prove
from this assumption that, for most reasonable quantum systems, if the
initial state is undecayed in the sense that \(\mathrm{Tr}(\rho E)=1\), then the
probability that the atom will decay in any given time interval \([0,t]\)
is equal to zero. That is, continual monitoring implies that the atom
will never decay.

These ideas have given rise to a large literature of responses. To
give a sampling: Ghirardi et al. (1979) and Pati (1996) have objected
that this Zeno-like model of a quantum measurement runs afoul of other
properties of quantum theory, such as the time-energy uncertainty
relations, which they argue should prevent the measurements in the
supertask sequence above from being made with arbitrarily high
frequency. Bokulich (2003) has responded that, nevertheless, such a
supertask can still be carried out when the measurement commutes with
the unitary evolution, such as when \(E\) is a projection onto an energy
eigenstate.

2. Supertasks in Relativistic Spacetime

In Newtonian physics, time passes at the same rate for all observers.
If Alice and Bob are both present at Alice’s 20th and 21st birthday
parties, both people will experience an elapsed time of one year
between the two events. (This is true no matter what Alice or Bob do
or where Alice and Bob go in between the two events.) Things aren’t so
simple in relativistic physics. Elapsed time between events is
relative to the path through spacetime a person takes between them. It
turns out that this fact opens up the possibility of a new type of
supertask. Let’s investigate this possibility in a bit more
detail.

2.1 Time in Relativistic Spacetime

A model of general relativity, a spacetime, is a pair \((M,g)\).
It represents a possible universe compatible with the theory. Here, \(M\)
is a manifold of events. It gives the shape of the universe. (Lots of
two-dimensional manifolds are familiar to us: the plane, the sphere,
the torus, etc.) Each point on \(M\) represents a localized event in space
and time. A supernova explosion (properly idealized) is an event. A
first kiss (properly idealized) is also an event. So is the moon
landing. But July 20, 1969 is not an event. And the moon is not an
event.

Manifolds are great for representing events. But the metric \(g\) dictates
how these events are related. Is it possible for a person to travel
from this event to that one? If so, how much elapsed time does a
person record between them? The metric \(g\) tells us. At each event, \(g\)
assigns a double cone structure. The cone structures can change from
event to event; we only require that they do so smoothly. Usually, one
works with models of general relativity in which one can label the two
lobes of each double cone as “past” and
“future” in a way which involves no discontinuities. We
will do so in what follows. (See figure 2.1.1.)

Fig 2.1.1. Events in spacetime and the associated double cones.

Intuitively, the double cone structure at an event demarcates the
speed of light. Trajectories through spacetime which thread the inside
of the future lobes of these “light cones” are possible
routes in which travel stays below the speed of light. Such a
trajectory is a worldline and, in principle, can be traversed
by a person. Now, some events cannot be connected by a worldline. But
if two events can be connected by a worldline, there is an
infinite number of worldlines which connect them.

Each worldline has a “length” as measured by the metric \(g\);
this length is the elapsed time along the worldline. Take two events
on a manifold \(M\) which can be connected by a worldline. The elapsed
time between the events might be large along one worldline and small
along another. Intuitively, if a worldline is such that it stays close
to the boundaries of the cone structures (i.e. if the trajectory stays
“close to the speed of light”), then the elapsed time is
relatively small. (See Figure 2.1.2.) In fact, it turns out that if
two events can be connected by a worldline, then for any number \(t>0\),
there is a worldline connecting the events with an elapsed time less
than \(t\)!

Fig 2.1.2. Elapsed time is worldline dependent.

2.2 Malament-Hogarth Spacetimes

The fact that, in relativistic physics, elapsed time is relative to
worldlines suggests a new type of bifurcated supertask. The idea is
simple. (A version of the following idea is given in Pitowsky 1990.)
Two people, Alice and Bob, meet at an event \(p\) (the start of the
supertask). Alice then follows a worldline with a finite elapsed time
which ends at a given event \(q\) (the end of the supertask). On the other
hand, Bob goes another way; he follows a worldline with an infinite
elapsed time. Bob can use this infinite elapsed time to carry out a
computation which need not halt after finitely many steps. Bob might
check all possible counterexamples to Goldbach’s conjecture, for
example. (Goldbach’s conjecture is the statement that every even
integer n which is greater than 2 can be expressed as the sum
of two primes. It is presently unknown whether the conjecture is
true. One could settle it by sequentially checking to see if each
instantiated statement is true
for \(n=4\), \(n=6\), \(n=8\), \(n=10\), and so on.) If
the computation halts, then Bob sends a signal to Alice at \(q\) saying as
much. If the computation fails to halt, no such signal is sent. The
upshot is that Alice, after a finite amount of elapsed time, knows the
result of the potentially infinite computation at \(q\).

Let’s work a bit more to make the idea precise. We say that a
half-curve is a worldline which starts at some event and is
extended as far as possible in the future direction. Next, the
observational past of an event q, OP(q), is
the collection of all events x such that there a is a worldline
which starts at x and ends at q. Intuitively, a (slower
than light) signal may be sent from an event x to an
event q if and only if x is in the
set OP(q). (See figure 2.2.1.)

Fig 2.2.1.
The observational past of an event and a half-curve. A signal can be
sent to \(q\) from every point in \(OP(q)\). No signal
can be sent to \(q\) from any point on the half-curve \(\gamma\).

We are now ready to define the class of models of general relativity
which allow for the type of bifurcated supertask mentioned above
(Hogarth 1992, 1994).

Definition. A spacetime \((M,g)\) is Malament-Hogarth if there is
an event \(q\) in \(M\) and a half-curve \(\gamma\) in \(M\) with infinite elapsed
time such that \(\gamma\) is contained in \(OP(q)\).

One can see how the definition corresponds to the story above. Bob
travels along the half-curve \(\gamma\) and records an infinite elapsed
time. Moreover, at any event on Bob’s worldline, Bob can send a signal
to the event \(q\) where Alice finds the result of the computation; this
follows from the fact that \(\gamma\) is contained in \(OP(q)\). Note that
Alice’s worldline and the starting point \(p\) mentioned in the story did
not make it to the definition; they simply weren’t needed. The half
curve \(\gamma\) must start at some event – this event is our starting
point \(p\). Since \(p\) is in \(OP(q)\), there is a worldline from \(p\) to \(q\). Take
this to be Alice’s worldline. One can show that this worldline must
have a finite elapsed time.

Is there a spacetime which satisfies the definition? Yes. Let \(M\) be the
two-dimensional plane in standard \(t,x\) coordinates. Let the metric \(g\)
be such that the light cones are oriented in the \(t\) direction and open
up as the absolute value of \(x\) approaches infinity. The resulting
spacetime (Anti-de Sitter spacetime) is Malament-Hogarth (see Figure
2.2.2).

Fig 2.2.2.
Anti-de Sitter Spacetime is Malament-Hogarth. A signal can
be sent to \(q\) from every point on the half-curve \(\gamma\).

2.3 How Reasonable Are Malament-Hogarth Spacetimes?

In the previous section, we showed the existence of models of general
relativity which seem to allow for a type of bifurcated supertask.
Here, we ask: Are these models “physically reasonable”?
Earman and Norton (1993, 1996) and Etesi and Németi (2002) have
articulated a number of potential physical problems concerning
Malament-Hogarth spacetimes. First of all, we would like Bob’s
worldline to be reasonablly traversable. In the Anti-de Sitter model
above, the half-curve \(\gamma\) has an infinite total acceleration. Bob
would need an infinite amount of fuel to traverse it! (Malament
1985)

Another problem for the Anti-de Sitter spacetime is that a
“divergent blueshift” phenomenon occurs. Intuitively, the
frequency of any signal Bob sends to Alice is amplified more and more
as he goes along. Eventually, even the slightest thermal noise will be
amplified to such an extent that communication is all but impossible.
So, if the counterexample to Goldbach’s conjecture comes late in the
game (or not at all), it is not clear that Alice can ever know this.

One can find Malament-Hogarth spacetimes which can escape both of the
problems mentioned above. Let \(M\) be a two-dimensional plane in
standard \(t, x\) coordinates which is then “rolled up” along
the \(t\) axis. Let the metric \(g\) be such that the light cones are oriented
in the \(t\) direction and do not change from point to point. (See Figure
2.3.1.)

Fig 2.3.1.
An acausal Malament-Hogarth spacetime.

Because worldliness can wrap around and around the cylinder, \(OP(q)=M\)
for any event \(q\). This allows for great freedom in choosing Bob’s
worldline \(\gamma\). In fact, we can choose it so that the total
acceleration is zero – no fuel is needed to traverse it. Moreover, we
can choose it so that there is also no divergent blueshift phenomenon
(see Earman and Norton 1993). But, alas, we have a new problem: the
spacetime is acausal. A worldline can start and end at the same event
allowing for a type of “time travel”. It is unclear if
spacetimes allowing for time travel are physically reasonable (see
Smeenk and Wüthrich 2011). It turns out that more complicated
examples can be constructed which avoid all the potential problems
mentioned so far and more (Manchak 2010). But such examples contain
spacetime “holes” which may not be physically reasonable
(see Manchak 2009). More work is needed to see if such problems can
also be overcome.

We conclude with one final potential problem which threatens to render
all Malament-Hogarth spacetimes physically unreasonable. Penrose
(1979) has conjectured that all physically reasonable spacetimes are
free of a certain type of “naked singularities” and the
breakdown of determinism they bring. Whether Penrose’s conjecture is
true or not is the subject of much debate (Earman 1995). But it turns
out that every Malament-Hogarth spacetime harbors these naked
singularities (Hogarth 1992). In sum, it is still an open question
whether Malament-Hogarth spacetimes are simply artifact of the
formalism of general relativity or if the kind of bifurcated supertask
they suggest can be implemented in our own universe.

Bibliography

Atkinson, D., 2007, “Losing energy in classical, relativistic and
quantum mechanics”, Studies in History and Philosophy of Modern
Physics, 38(1): 170–180.